We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.
@InProceedings{colindeverdiere_et_al:LIPIcs.ESA.2021.32, author = {Colin de Verdi\`{e}re, \'{E}ric and Magnard, Thomas}, title = {{An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.32}, URN = {urn:nbn:de:0030-drops-146139}, doi = {10.4230/LIPIcs.ESA.2021.32}, annote = {Keywords: computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability} }
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